UC-NRLF 


3M    437 


' 

A  STUDY 

OF 

THE  LIGHT  CURVE  OF  THE  VARIABLE  STAR  U  PEGASI 


BASED   ON 


THE  OBSERVATIONS  OF  HARVARD  COLLEGE  OBSERVATORY  CIRCULAR  NO.  23. 


BULLETIN  NO.  I. 


OP 


THE  ASTRONOMICAL   OBSERVATORY 


OF  THE 


UNIVERSITY  OF  ILLINOIS, 

UKBANA,  ILLINOIS. 


BY 

G.  W.  I  MYERS,  DIRECTOR. 


CAMBRIDGE,  U.S.A.: 
PRINTED   FOR   THE   UNIVERSITY  OF   ILLINOIS, 

at  tljc  JSmbrvsitB 
1898. 


ASTRONOMY  LIBRARY 


A   STUDY   OF   THE   LIGHT   CURVE 


OP 


THE  VARIABLE   STAR   U  PEGASI. 


PROFESSOR  PICKERING  has  shown  in  Harvard  College  Observatory  Circular  No.  23,  that 
U  Pegasi  no  longer  deserves  the  distinction  of  being  considered  the  variable  of  shortest  known 
period.  Contrary  to  the  usual  form  of  contestant,  in  the  present  instance,  the  disputant  for 
pre-eminence  in  this  particular  is  not  a  newly  discovered  variable  of  shorter  period  than  any 
hitherto  known,  but  is  the  variable  a>  Centauri  19,  discovered  by  Baily  some  time  since  and  found 
to  have  the  period  7A  llm.  Manifestly,  therefore,  U  Pegasi,  whose  period  has  until  recently  been 
regarded  as  lying  between  3\0  and  5h.6,  has  been  turned  down  the  list,  not  because  of  the  exces- 
sive shortness  of  the  period  of  some  other  star.  The  reason  for  the  change  lies  in  the  fact  that 
the  inequality  of  brightness  of  the  alternate  minima  of  U  Pegasi  escaped  detection,  until 
Professor  Pickering's  discussion  revealed  it  last  winter.  His  observations,  published  in  the  form 
of  a  light  curve  and  reproduced  in  substance  in  Plate  I.  accompanying  this  paper,  showed 
the  most  probable  period  based  upon  all  preceding  observations  to  be  about  4A.5 ;  but  that,  in 
view  of  the  failure  of  former  observers  to  recognize  the  difference  of  brightness  of  the  minima, 
this  period  should  be  doubled.  Applying  a  slight  correction  to  the  double  value,  shown  to  be 
justified  by  more  recent  observations,  he  states,  as  the  best  value  for  the  period-length  of  this 
star  8*  59m  41*.  The  mean  value  of  the  brightness  at  the  two  approximately  equal  maxima 
is  9™.30 ;  at  the  secondary  minimum,  the  brightness  is  9m.75,  and  at  the  primary  it  is  9m.90. 
The  plate  referred  to  gives  the  observations  on  such  a  scale  that  one  division  in  the  ordinates 
corresponds  to  0.1  magnitude  and,  in  the  abscissas,  to  half  an  hour.  The  above  mentioned 
circular  states  that  the  total  number  of  settings  here  represented  is  2784  and  that  the  time  of 
observation,  including  rests,  is  30  hours.  Each  dot  in  the  plate  represents  80  settings,  the  dots 
being  formed  by  the  method  of  overlapping  means. 

The  least  difference  of  stellar  brightness  of  whose  existence  the  eye  can  be  certain,  being 
about  0.1  of  a  magnitude,  and  the  difference  of  brightness  between  the  primary  and  secondary 
minima,  as  stated  in  the  Circular,  lying  so  near  this  limit,  i.  e.  =  0.15  of  a  magnitude,  there  would 
seem  to  be  just  cause  for  suspicion  that  this  apparent  difference  has  arisen  from  the  rather  large 
accidental  errors  always  attaching  to  photometric  observations.  In  view  of  the  almost  uniformly 
high  degree  of  excellence  attained  in  the  past  by  Professor  Pickering's  forms  of  photometer, 
it  cannot  be  denied  that  the  results  of  photometric  measures  are  on  the  whole  to  be  ascribed  a 
far  higher  measure  of  accuracy  than  belongs  to  photometric  estimates.  A  recent  personal  study 
of  /3  Lyrae's  light  variation  made  with  one  of  Professor  Pickering's  polarization  photometers 
removes  from  the  writer's  mind  the  last  vestige  of  doubt  as  to  the  certainty  of  the  existence  of 
this  difference  of  brightness  at  the  minima.  But  whatever  doubt  may  have  existed  for  a  time  as 
to  its  reality,  it  would  seem  that  the  following  statements  of  Professor  Pickering  in  the  Ap.  J. 
for  March  of  this  year,  ought  to  dispel  it  quite  effectually.  "Twelve  observations,  each  consisting 
of  sixteen  settings,  were  made  when  the  star  was  within  twenty  minutes  of  its  primary  minimum. 


ffi'753275 


A   STUDY    OF   THE    LIGHT    CUUVE 


Deriving  from  each  of  these,  by  means  of  the  light  curve,  the  magnitude  of  this  minimum, 
we  obtain  on  Oct.  18,  1897,  9.89,  9.94,  and  9.96;  on  Dec.  30,  9.90,  9.95,  and  9.93;  on 
Jan.  1,  1898,  9.93,  9.86,  and  9.85  ;  on  Jan.  5,  9.85,  and  on  Jan.  7,  9.86  and  9.88.  Mean  of 
all  =  9.90  ;  greatest  value  =  9.96 ;  least  value  =  9.85  and  average  deviation  =  ±  0.035.  Similarly, 
fourteen  observations  were  taken  within  twenty  minutes  of  the  secondary  minimum  with  the 
results  on  Oct.  18,  1897,  9.75  and  9.71 ;  on  Oct.  29,  9.74,  9.69,  9.70  and  9.70 ;  on  Dec.  28, 
9.78,  9.77,  9.76  and  9.80;  on  Jan.  3,  1898,  9.77,  9.77,  9.74  and  9.78.  Mean  of  all  =  9.75 ; 
greatest  value  =  9.80 ;  least  value  =  9.69,  and  average  deviation  =.  ±  0.029."  The  probable  errors 
would,  of  course,  be  smaller  than  the  "  average  deviations."  Obviously,  average  deviations, 
probable  errors,  and  the  like,  mean  nothing  at  all  here,  or  they  mean  that  an  error  in  the  great- 
est value  of  the  primary  minimum  large  enough  to  make  it  equal  to  even  the  least  value  at  the 
secondary  cannot  be  entertained  as  a  probability,  since  it  would  mean  the  commission  of  a 
systematic  error  nearly  twice  as  great  as  the  average  deviation  and  more  than  twice  as  great  as 
the  probable  error.  The  chances  against  this  would  be  a  little  worse  than  1  to  5.2.  The  inter- 
nal evidence  of  the  observations  is,  it  would  seem,  quite  conclusive  in  favor  of  the  reality  of  the 
discrepancy.  The  statements  just  quoted  show,  moreover,  that  especial  attention  was  directed 
to  the  point  in  question,  and  it  seems  therefore  scarcely  reasonable  to  suspect  that,  under  such 
circumstances,  an  error  of  0.15  of  a  magnitude  could  elude  certain  detection  and  confirmation. 

Assuming  the  reality  of  this  difference,  the  light  curve  appears  to  be  susceptible  of  treatment 
by  essentially  the  same  method  as  that  adapted  and  used  by  the  writer  in  his  recent  discussion 
of  Beta  Lyrae's  light  curve  entitled :  UNTERSUCHUNGEN  DEBEB  DEN  LICHTWECHSEL  DBS  STERNES 
/3  LYRAE,  Muenchen,  1896.  It  is  the  purpose  of  this  Bulletin  to  present  the  results  and  an  out- 
line of  the  method  used  in  a  recent  study  of  U  Pegasi,  based  essentially  upon  the  observations 
of  Pickering's  Circular  No.  23,  and  by  the  method  'more  fully  developed  in  the  foregoing  disser- 
tation. The  fundamental  hypothesis  underlying  the  whole  discussion  is  that  the  light  curve  of 
U  Pegasi  is  capable  of  being  explained  on  the  satellite  theory. 

ECCENTRICITY. 

The  uncertainty  in  the  instants  of  maximum  brightness  as  indicated  by  the  light  curve  of 
Plate  I.,  obviously  precludes  the  possibility  of  deriving  an  approximate  value  of  the  orbital 

eccentricity  of  the  component  from  the 
chief  epochs  of  light  variation,  as  was  done 
with  /3  Lyrae.  One  may  readily  convince 
himself  by  considerations  adduced  below, 
however,  that  this  eccentricity  must  be 
quite  small. 

Assuming  the  light  fluctuations  to  be 
due  to  the  mutual  eclipses  of  two  unequally 
bright  bodies,  we  should  have  the  chief 
epochs  occurring  when  the  relative  posi- 
tions of  the  components  are  as  indicated  in 
the  subjoined  figure.  That  the  bodies  are 
unequally  bright,  follows  at  once  from  the 
consideration  that  at  Min.  I.  the  brightness 
of  the  star  is  reduced  by  41  per  cent  of  its 
maximum  brightness,  and  at  Min.  II.  by 
only  31  per  cent ;  unless  the  orbital  eccen- 
tricity is  assumed  quite  large.  It  will  now  be  shown  that  the  latter  cannot  be  the  case. 
Assuming  also  provisionally,  that  both  bodies  are  spheres,  a  lower  limit  for  the  eclipse- 


tVIHIMUM        Jl. 


OF  THE  VARIABLE  STAR  U  PEGASI.  3 

duration  at  Min.  I.  can  bo  easily  obtained  from  the  observational  curve  given  in  Fig.  1.  A  little 
reflection  will  make  it  clear  tbat  the  shorter  the  eclipse-duration  be  taken,  the  larger  will  be  the 
corresponding  distance  between  centres  of  the  components.  If,  then,  we  assume  that  the  eclipse 
has  not  begun  until  the  light  curve  has  fallen  quite  appreciably  and  that  it  has  ended  shortly 
before  the  curve  ceases  to  rise,  we  shall  obtain  a  value  for  the  duration  of  the  eclipse,  at  all 
events  short  enough,  —  perhaps  too  short,  —  and  the  corresponding  value  of  the  distance  of  cen- 
tres must  be  at  all  events  great  enough  —  perhaps  too  great.  Proceeding  thus,  I  obtain  3*.3  for 
the  interval  shorter  than  which  the  eclipse-duration  at  Min.  I.  cannot  be.  The  corresponding 
value  of  the  distance  between  centres  may  then  be  regarded  as  fixing  a  superior  limit  for  this 
orbital  element. 

Calling  the  radius  of  the  larger  component  unity  and  of  the  smaller  «,  the  radius  vector  of 
the  true  orbit,  r,  one-half  the  distance  between  the  nearest  points  of  the  positions  of  the  com- 
panions at  the  beginning  and  end  of  the  eclipse,  x,  and  for  this  roughly  approximate  purpose, 
assuming  e  to  be  zero,  we  have  from  the  figure : 

CPC'  >  3.3  fji  =  132°       (p  =  Zir/P  =  360°/9  =  40°) 
Hence, 

CP  D  >  66°  and  r  <  (x  +  K)  esc  66° 

<  1.0946  (x  +  K) 


But  since  x  ^  1  and  K  <  1,  we  shall  have  r  ^  2.189  times  the  radius  of  the  larger  com- 
panion. So  small  a  distance  of  centres  relative  to  the  dimensions  of  the  primary,  coupled 
with  a  large  orbital  eccentricity,  would  be  highly  improbable  theoretically  in  any  case,  and 
assuming  distinct  duplicity,  would  be  a  physical  impossibility  on  any  other  hypothesis  than 
that  the  extent  of  the  secondary  is  quite  inconsiderable  compared  with  that  of  the  primary. 
The  approximately  equal  fall  of  brightness  at  the  minima,  together  with  the  similarity  of  form 
of  the  light  curve  in  the  neighborhood  of  these  two  chief  epochs,  argues  strongly  for  the  view 
that  the  form  and  dimensions  of  the  companions  cannot  be  widely  different,  and  this  latter  view 
is  still  further  supported  by  the  fact  that  the  relative  brightness  of  the  components  is  found 
later,  independently  of  any  hypothesis  regarding  the  ratio  of  the  radii,  to  be  about  0.8. 

It  may  therefore  be  assumed  as  a  first  approximation  that  e  =  0,  and  we  shall  now  proceed 
to  determine  the  value  of  the  ratio  of  the  brightness  of  the  companions  and  to  fix  the  limits 
within  which  the  ratio  of  the  radii  must  be  comprised.  We  shall  then  undertake  to  find  the 
most  probable  value  of  this  latter  ratio  by  direct  reference  to  the  light  curve  of  the  star- 

CIRCULAR  ORBITAL  ELEMENTS  AND  LIGHT  RATIO  OF  THE  COMPONENTS  OF  U  PEGASI. 

The  chief  epochs  of  the  light  curve  shall  be  designated  in  order  from  left  to  right  in  Figure  1 
as  Min.  I.,  Max.  I.,  Min.  II.  and  Max.  II.  From  the  curve  Max.  I.  is  seen  to  have  a  brightness 
of  9.32  magnitude  and  Max.  II.  of  9.34  magnitude,  so  that  the  mean  value  9m.33  has  been  used 
throughout  the  discussion  for  the  brightness  at  both  the  maxima.  For  the  brightness  at  Min.  I., 
the  value  9.90  magnitude  has  been  used  and  for  Min.  II.,  9.75  magnitude.  Reducing  these 
differences  in  stellar  magnitudes  at  the  chief  epochs  of  variability  to  their  equivalent  light 
ratios,  by  the  aid  of  Pogson's  scale,  we  obtain: 

Brightness  at  Min.  II. 

Brightness  at  Min.  I. 

Brightness  at  Mean  Max. 

Brightness  at  Min.  1. 


A   STUDY    OF   THE    LIGHT    CURVE 


Ectaining  the  nomenclature  of  the  foregoing  paragraph,  calling  the  light  ratio  of  the  com- 
ponents X  and  the  portion  of  the  discs  common  to  both  bodies  at  the  middle  of  the  eclipses  a, 
the  preceding  equations  give  the  following  : 

1  +  «2  X  -  a  K2  X 
(1)  -  =  c 


K 

(2)  -  =  m. 

2 


If  it  be  thought  desirable  to  include  the  possibility  of  a  flattening  of  the  discs,  we  may 
assume,  as  a  means  of  making  a  first  approximation  to  the  general  effect  of  such  deformation, 
that  the  bodies  are  similar  ellipsoids  of  revolution  and  designate  by  q,  the  common  ratio  of  the 
semi-major  to  the  semi-minor  axis,  whereupon  equation  (2)  must  be  replaced  by 

i  j.  if  \ 

/n     \  '     K    ** 

(2a)         q —  =  m 

(Conf.  Ap.  J.  Vol.  VII.,  p.  13,  where  a  «2  should  be  stricken  from  the  numerator  of  (e).) 
From  (1)  and  (2  a)  we  find  readily 

aK2X 

and  a  K!! 

whence,  dividing,  we  get 

(5)  X  =  (m-cq)/(m-q). 

Neglecting  the  flattening  provisionally,  i.e.,  putting  q  —  1,  (5)  gives,  when  the  foregoing 
values  of  c  and  m  are  substituted, 

X  =  0.7865. 
From  (3)  and  (1),  we  obtain 

m  m —  co 


.  a- 


K*       m  —  q         m  —  q 
and  (4)  gives 


a 
m  —  q  =  m  — 


K2X 


Since  now,  a  K  2  and  1  +  K  2  X  are  essentially  positive,  being  quantities  of  light,  this  latter 
relation  shows  that  m  must  be  greater  than  q.     Consequently, 


•G) 


m 


da         m  —  q 

is  also  a  positive  magnitude.     (l//c2)  and  a  therefore,  increase  and  decrease  together,  so  that 
the  maximum  value  of  a  corresponds  to  the  maximum  value  of  (l/«2). 
If  now, 

K2  ^  1,  then  a  <  1  (from  geometrical  considerations), 
and  it  follows  from  (6)  that, 


,  or  K*  > 


K"        m  —  q  c  q 

But  if, 

K"  >  1,  we  may  put  a  <  —  (also  for  geometrical  reasons), 


OF   THE   VARIABLE    STAR   U    PEGASI. 


We  have  from  (6), 


and  hence, 


m 


m  —  q 


m  —  cq 


m  —  cq 
Summing  up  both  contingencies  into  a  single  condition,  there  results  : 

(?) 


From  this,  we  have  also, 
and  finally, 


cq 


m  —  cq 


cq 


m  —  cq 


m 


-*--  \d 


Substituting  now  the  former  values  of  m  and  c,  we  obtain 

q  >  0.787. 

It  does  not  therefore  appear  to  be  necessary  to  assume  the  existence  of  a  flattening  for  U 
Pegasi,  such  as  was  shown  to  be  necessary  in  my  Dissertation  on  Beta  Lyrae,  p.  30,  for  the 
latter  star. 

Taking  again  the  value  of  q  as  unity,  and  substituting  in  (7)  we  find  : 

0.6014  <  KS  <  1.845,    or    0.7755  <  K  <  1.358. 

The  following  test  values  distributed  linearly  over  this  interval  were,  therefore,  selected  for 
criteria  to  an  approximation  to  «  : 

0.80,        0.85,        1.00,        1.15         and        1.35, 

and  for  each  of  these  values  a  light  curve  was  computed  by  the  method  and  with  the  results 
given  below. 

Using  the  portion  of  the  light  curve  lying  within  1.5  hours  before  and  after  Min.  I.,  and  the 
notation  (v,  Fig.  3)  and  equations  developed  in  my  dissertation  and  published  in  the  Ap.  J.  for 
Jan.,  1898,  1  have  to  compute  the  values  of  M  and  H  from  the  data  furnished  by  the  Hght 
curve  and  then  for  K.  <  1,  to  solve  the  transcendental  equations  : 


and  for  «  >  1, 


M  =  <f>  +  K2  <{>"  -  K  sin  (<£"  + 
H  =  K?  <t>'  —  <f>  +  K  sin  (<#>'  — 


M  =  </>  +  K 

H  =    <f>!    - 


-  K  sn  ( 

"    +  K  SlU 


for  <£  and  <£"  and  when  K  =  1, 

(9  a)  M  —  2  <£  —  K  sin  2  <£  =  2  <£  —  sin  2  <£  (H  being  here  zero). 

These  solutions  may  be  made  most  conveniently  by  means  of  tables  giving  the  values  of  M 
and  H  for  suitably  chosen  values  of  <£  and  <j>',  from  which  approximate  values  of  <£  and  <£'  may 
be  interpolated,  which  may  then  be  corrected  by  the  following  differential  formulae  : 


(10) 


S<i  = 


2«tg. 


sn 


+ 


and  S  <£  = 


2  K  tg  $'  (sin  <£'  —  i 


- ,  f  or  K  <  1. 


A   STUDY    OF   THE  LIGHT    CURVE 


When  K  >  1,  we  shall  have  to  use  instead  of  the  latter, 

8/7 


(11) 


2  K  tg  </>'  sin  ($  —  <£') 

If  it  be  desired  to  assume  a  value  of  q  a  little  greater  than  unity,  it  will  then  be  necessary  to 
compute  X  from  equation  (5)  above.     Differentiating  (5)  with  respect  to  q,  I  obtain 

d  A.  _        in  (1  +  c) 

~»      =         ~,  77' 

(I  q  (m  —  qY 

an  essentially  negative  magnitude. 

X  and  q  therefore  change  against  each  other,  so  that  an  increase  in  q  will  necessitate  a  decrease 
in  X.     Again  designating  the  maximum  value  of  K2  by  K2  and  the  minimum  by  k2  we  have 


m  —  q 
cq 


and    Ki  = 


m  —  c  q 


Differentiating  these  with  respect  to  q,  we  find, 

Ct  rC  VYIi 

dq  c  q2 


and 


d  q 


=  + 


m 


(m 


The  former  of  these  differential  coefficients  is  essentially  negative,  and  the  latter  is  essen- 
tially positive.  An  augmentation  of  q  will  therefore  depress  the  minor  and  elevate  the  major 
limit  of  K2;  and  to  be  able  to  include  a  value  of  q  somewhat  larger  than  unity,  values  of  M  and 
If  were  also  computed  for  K  =  0.70.  The  table  of  computed  M's  and  H's  is  given  here. 

AUXILIARY  TABLES  FOR  INTERPOLATING  APPROXIMATE   VALUES   OF  <f>  AND  </>". 


<(>  for  K  <1 
or 
£"for/c>l 

K  =  0.70 

K  =  0.80 

K  =  0.85 

K=  1.15 

K  =  1.35 

M 

H 

M 

n 

M 

n 

M 

H 

M 

n 

o   / 

0  00 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

2  00 

0.0001 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0001 

0.0000 

0.0000 

0.0000 

4  00 

0.0005 

0.0001 

0.0006 

0.0001 

0.0005 

0.0003 

0.0007 

0.0001 

0.0009 

0.0001 

6  00 

0.0018 

0.0007 

0.0017 

0.0005 

0.0017 

0.0003 

0.0022 

0.0003 

0.0031 

0.0010 

8  00 

0.0043 

0.0009 

0.0040 

0.0006 

0.0038 

0.0005 

0.0051 

0.0007 

0.0077 

0.0014 

10  00 

0.0086 

0.0013 

0.0080 

0.0009 

0.0077 

0.0008 

0.0100 

0.0008 

0.0152 

0.0021 

12  00 

0.0148 

0.0029 

0.0137 

0.0017 

0.0133 

0.0014 

0.0171 

0.0013 

0.0261 

0.0041 

14  00 

0.0319 

0.0044 

0.0216 

0.0025 

0.0210 

0.0018 

.  0.0274 

0.0019 

0.0414 

O.OOG8 

16  00 

0.0353 

0.0065 

0.0325 

0.0037 

0.0314 

0.0026 

0.0405 

0.0026 

0.0618 

0.0090 

18  00 

0.0503 

0.0093 

0.0461 

0.0054 

0.0444 

0.0038 

0.0577 

0.0038 

0.0880 

0.0148 

20  00 

0.0690 

0.0136 

0.0631 

0.0076 

0.0610 

0.0056 

0.0792 

0.0062 

0.1207 

0.0201 

22  00 

0.0933 

0.0202 

0.0838 

0.0104 

0.0805 

0.0072 

0.1050 

0.0082 

0.1607 

0.0274 

24  00 

0.1195 

0.0247 

0.1087 

0.0142 

0.1043 

0.0096 

0.1306 

0.0107 

0.2086 

0.0300 

26  00 

0.1522 

0.0328 

0.1378 

0.0184 

0.1320 

0.0126 

0.1735 

0.0129 

0.2648 

0.0473 

28  00 

0.1908 

0.0422 

0.1715 

0.0231 

0.1644 

0.0161 

0.2137 

0.0175 

0.3112 

0.0613 

30  00 

0.2344 

0.0542 

0.2104 

0.0293 

0.2014 

0.0280 

0.2020 

0.0228 

0.4079 

0.0779 

32  00 

0.2876 

0.0693 

0.2549 

0.0367 

0.2433 

0.0251 

0.3163 

0.0281 

0.4959 

0.0983 

34  00 

0.3307 

0.0710 

0.3053 

0.0458 

0.2906 

0.0309 

0.4022 

0.0343 

0.5969 

0.1240 

36  00 

0.4176 

0.1120 

0.3621 

0.0555 

0.3436 

0.0381 

0.4457 

0.0425 

0.7116 

0.1548 

38  00 

0.4998 

0.1438 

0.4055 

0.0696 

0.4024 

0.0465 

0.5222 

0.0518 

0.8447 

0.1972 

40  00 

0.5977 

0.1863 

0.4968 

0.0853 

0.4679 

0.0569 

0.6059 

0.0022 

0.9942 

0.2446 

OF  THE  VARIABLE  STAR  U  PEGASI. 


<(>  for  K  <1 
or 
<f,"  for  K  >1 

K  —  0.70 

K  =  0.80 

K  =  0.85 

K  =  l.K 

K  —  1.36 

M 

H 

M 

H 

M 

H 

M 

H 

M 

B 

o   / 

42  00 

0.7218 

0.2503 

0.5772 

0.1047 

0.5397 

0.0681 

0.6984 

0.0752 

1.1765 

0.3103 

43  00 

0.8028 

0.2994 

— 

— 

— 

— 

— 

— 

1.2694 

0.3591 

43  30 

0.8460 

0.3328 

— 

— 

— 

— 

— 

— 

— 

— 

44  00 

0.9281 

0.3818 

0.6658 

0.1296 

0.6162 

0.0858 

0.7996 

0.0908 

1.3790 

0.4010 

44  20 

0.9797 

0.4388 

— 

— 

— 

— 

— 

— 

— 

— 

44  26 

1.0297 

0.4783 

— 

— 

— 

— 

— 

— 

— 

— 

45  00 

— 

— 

— 

— 

— 

— 

— 

— 

1.5036 

0.4638 

46  00 

— 

— 

0.7666 

0.1600 

0.7066 

0.1002 

0.9103 

0.1087 

1.6499 

0.5445 

47  00 

— 

— 

0.8229 

0.1731 

— 

— 

— 

— 

1.8417 

0.6700 

47  30 

— 

— 

— 

— 

— 

— 

— 

— 

1.9983 

0.7763 

47  48 

— 

— 

— 

— 

— 

— 

— 

— 

2.1843 

0.9573 

48  00 

— 

— 

0.8810 

0.1999 

0.8028 

0.1217 

1.0308 

0.1303 

— 

— 

49  00 

— 

— 

0.9488 

0.2288 

— 

— 

— 

— 

— 

— 

50  00 

— 

— 

1.0215 

0.2009 

0.9094 

0.1490 

1.1641 

0.1578 

_ 

— 

51  00 

— 

__ 

1.1055 

0.3032 

— 



— 

— 

— 

— 

51  30 

— 



1.1534 

0.3301 

— 

. 

— 

— 



— 

52  00 

— 

_ 

1.2082 

0.3632 

1.0290 

0.1842 

1.3129 

0.1929 

— 

— 

52  30 

— 

— 

1.2726 

0.4083 

— 

— 

— 

— 

— 

— 

53  00 

— 

— 

1.3736 

0.4847 

— 

— 

— 

— 

— 

— 

53  8.7 

— 

— 

1.4531 

0.5575 

— 

— 

— 

— 

— 

— 

54  00 

— 

— 

— 

— 

1.1675 

0.2480 

1.4749 

0.2376 

— 

— 

56  00 

— 

— 

— 

— 

1.3320 

0.3044 

1.6574 

0.2991 

— 

— 

57  00 

— 

— 

— 

— 

1.4389 

0.3728 

— 

— 

— 

— 

57  30 

— 

— 

— 

— 

1.5066 

0.4060 

— 

— 

— 

— 

58  00 

— 

— 

— 

— 

1.5992 

0.4734 

1.8764 

0.3874 

— 

— 

58  12.7 

— 

— 

— 

— 

1.7040 

0.5666 

— 

— 

— 

— 

58  30 

— 

— 

— 

— 

— 

— 

1.9414 

0.4192 

_ 

— 

59  00 

— 

— 

— 

— 

— 

— 

2.0187 

0.4567 

— 

— 

59  30 

— 

— 

— 

— 

— 

— 

2.0970 

0.5050 

__ 

— 

CO  00 

— 

— 

— 

— 

— 

— 

2.2032 

0.5786 

— 

— 

60  25 

~~ 

— 

— 

— 

— 

— 

2.3976 

0.7439 

— 

— 

The  table  gives  the  values  of  M  and  ff,  of  course,  only  up  to  </>  —  sin  *,  or  to  <j>'  =  sin    -> 
according  as  «  ^  1. 

The  expression  for  AT  when  K  =  1  is  so  simple  as  to  render  the  use  of  an  auxiliary  table  un- 
necessary, and  this  case  has  therefore  not  been  included  in  the  foregoing  lists. 
M  and  H  are  connected  with  the  observations  by  means  of  the  relations  : 

(12)        M  =  TT  (1  +  K2  X)  (1  -  «/)    and    H  =  IT  Ka  -  *  (1  +  *2  A.)  (1  -  J), 

where  J  is  obtained  from  the  light  curve  by  subtracting  the  ordinate  of  the  curve  for  any 
given  instant  from  the  mean  ordinate  for  the  maxima,  calling  this  difference  A(r  and  substituting 
in  the  equation : 

log  J  =  0.04  A G         (\G  being  in  tenths  of  a  magnitude). 

The  values  of  M  and  H,  on  the  various  hypotheses  for  K  and  for  the  times  preceding  and 
following  Min.  I.  given  in  the  first  column,  are  tabulated  here. 


A    STUDY   OF   THE    LIGHT    CURVE 


VALUES  OF  M  AND  H  COMPUTED  FROM  THE  LIGHT  CURVE  FOR  THE  EPOCHS  t. 


t 

K  =  0.80 

K  =  0.85 

K  =  1.00 

K=115 

K  =  1.35 

Jt 

H 

M 

H 

M 

H 

M 

H 

M 

H 

h. 

-1.50 

0.1624 

1.8482 

0.1694 

2.1004 

0.1930 

2.9486 

0.2203 

3.9345 

0.2628 

5.4628 

-1.25 

0.3910 

1.6196 

0.4077 

1.8621 

0.4625 

2.6771 

0.5303 

3.6245 

0.6325 

5.0931 

-1.00 

0.6903 

1.3203 

0.7200 

1.5498 

0.8201 

2.3215 

0.9364 

3.2184 

1.1168 

4.G088 

-0.75 

1.0671 

0.9435 

1.1129 

0.1569 

1.2677 

1.8739 

1.4476 

2.7072 

1.7264 

3.9992 

-0.50 

1.4496 

0.5610 

1.5118 

0.7580 

1.7221 

1.4195 

1.9664 

2.1884 

2.3452 

3.3804 

-0.25 

1.8458 

0.1648 

1.9250 

0.3448 

2.1927 

0.9489 

2.5038 

2.6510 

2.9861 

2.7395 

-0.12| 

1.9067 

0.1039 

1.9885 

0.2813 

2.2650 

0.8766 

2.5864 

1.5684 

3.0847 

2.6409 

0.00 

1.9326 

0.0780 

2.0156 

0.2542 

2.2959 

0.8457 

2.6216 

1.5332 

3.1267 

2.5989 

+  0.12£ 

1.8986 

0.1120 

1.9801 

0.2897 

2.2555 

0.8861 

2.5755 

1.5793 

3.0727 

2.6529 

+  0.25 

1.7192 

0.2914 

1.7930 

0.4768 

2.0423 

1.0993 

2.3321 

1.8227 

2.7814 

2.9442 

+  0.50 

1.2380 

0.7726 

1.2912 

0.9786 

1.4708 

1.6708 

1.6794 

2.4754 

2.0030 

3.7226 

+  0.75 

0.8358 

1.1758 

0.8716 

1.3982 

0.9928 

2.1488 

1.1337 

3.0211 

1.3521 

4.3735 

+  1.00 

0.5383 

1.4623 

0.5614 

1.7084 

0.6395 

2.5021 

0.7302 

3.4246 

0.8708 

4.8548 

+  1.25 

0.2748 

1.7258 

0.2866 

1.9832 

0.3265 

2.8151 

0.3728 

3.7820 

0.4446 

5.2810 

+  1.50 

-0.0642 

1.9464 

0.0670 

2.2028 

0.0763 

3.0653 

0.0871 

4.0677 

0.1039 

5.6217 

(13) 


The  distance  of  centres,  r,  is  seen  from  the  accompanying  figure  to  be  given  by 

*  8ln    *"  ± 


p  = 


sin 


where  sin  0  =  K  sin  #>   and 


=  180°. 


The  figure  relates  only  to  the  case  in  which  K  <1  and  <f>"  <90°,  but  the  modifications 
necessary  to  adapt  it  to  the  cases  where  K  >1  and  </>"  <90°,  are  so  obvious,  that  they  may 
be  left  to  the  reader. 

Assuming   now  a   circular  orbit,  and   denoting   by   a   and   ft  +  -,   the   longitude   in   the 

2t 

apparent  orbit  and  the  true  anomaly  in   the  real  orbit  respectively,  both  counted  from  the 
node,  and  calling  r  and  p  the  radii  vectorcs  in  the  true  and  apparent  orbits,  we  may  write, 


p  cos  a  =  r  sin  ft  and   tg  a  =  cos  i  cot  /?, 
pa  =  r2  sin2  J3  +  r2  cos2  i  cos2  /?. 


whence, 

(14) 

Calling,  for  brevity, 

(14  a)  x  =  r2   and  y  =  r2  cos2  i  =  x  cos2  i, 

and  we  then  have  the  following  simple  relation  between  the  various  magnitudes ; 

(15)  p2  =  x  sin*  ft  +  y  cos2  ft.     (ft  =  p.  t  —  40°  t,  t  in  hours  from  Min.  I.), 

which  holds  for  all  cases  except  when  the  smaller  disc  is  projected  wholly  upon  the  larger 
at  the  epoch  of  Min.  I. 

The  solution  of  equations  (8),  (9)  and  (9a)  for  the  five  hypothetical  values  of  K  gave 
the  results  here  tabulated. 


OF   THE    VARIABLE    STAR   U   PEGASI. 


9 


TABULATED  VALUES  OF  <f>  AND  p. 


t 

K  =: 

D.80 

K  — 

D.85 

K  — 

LOO 

K  = 

1.15 

K  =z 

1.36 

<*> 

P 

* 

P 

* 

P 

W 

P 

$H 

P 

o     / 

0       / 

0       / 

0       / 

0      / 

-1.50 

27  30.3 

1.5403 

28  18.5 

1.5440 

30  40.0 

1.7202 

28  17.0 

1.8510 

25  56.0 

2.0210 

-1.25 

36  56.0 

1.3276 

38  10.4 

1.3697 

41  46.5 

1.4916 

38  12.5 

1.6066 

34  39.0 

1.7515 

-1.00 

44  30.5 

1.0987 

46  17.5 

1.1381 

51  25.5 

1.2472 

46  26.6 

1.3449 

41  15.0 

1.4710 

-0.75 

50  33.0 

0.8440 

53  16.0 

0.8813 

60  45.0 

0.9774 

53  43.0 

1.0557 

46  26.2 

1.1379 

-0.50 

53  8.0 

0.6012 

57  32.1 

0.6397 

68  43.0 

0.7260 

58  40.3 

0.7848 

36  18.0 

0.5558 

-0.25 

46  12.6 

0.3482 

56  44.0 

0.3951 

76  8.5 

0.4790 

45  51.0 

0.2360 

47  35.0 

0.8292 

-0.12 

41  52.0 

0.3038 

55  28.6 

6.3577 

77  14.0 

0.4420 

51  47.3 

0.2833 

47  46.2 

0.8805 

0.00 

39  8.0 

0.2841 

54  44.0 

0.3411 

77  42.0 

0.4260 

53  29.0 

0.3024 

47  47.2 

0.8897 

+  0.12 

42  40.0 

0.3101 

55  40.0 

0.3626 

77  5.5 

0.4474 

51  10.3 

0.2771 

47  45.4 

0.8726 

+  0.25 

50  45.2 

0.4321 

58  1.4 

0.4745 

73  50.0 

0.5568 

60  21.8 

0.6006 

46  28.8 

0.7254 

+  0.50 

52  13.0 

0.7342 

55  33.3 

0.7718 

64  26.3 

0.8630 

56  14.0 

0.9326 

47  33.3 

0.9982 

+  0.75 

47  13.6 

0.9969 

49  19.0 

1.0356 

55  16.1 

1.1394 

49  33.3 

1.2298 

43  45.5 

1.3331 

+  1.00 

41  4.3 

1.1781 

42  34.0 

1.2512 

46  54.4 

1.3664 

42  39.0 

1.4729 

38  23.0 

1.6037 

+  1.25 

32  49.0 

1.4288 

30  50.2 

1.4729 

36  51.0 

1.6004 

33  50.0 

1.7226 

30  52.0 

1.8803 

+  1.50 

20  40.0 

1.7093 

22  18.1 

1.8518 

20  40.0 

1.9909 

19  1.0 

2.2309 

A  comparison  of  the  values  of  p  on  the  last  two  hypotheses  for  K,  shows  at  once  that 
these  values  of  K  need  not  be  further  considered,  since  the  values  of  p  in  both  cases  fall 
for  a  time,  reach  a  minimum  before  Min.  I.,  rise  to  a  maximum  value  about  the  time  t  =  0, 
fall  to  a  second  minimum  value,  and  then  rise  continuously ;  and  since  p  denotes  the  radius 
vector  of  the  apparent  orbit,  which  latter  must  be  an  ellipse,  obviously  such  a  variation  of 
it  must  be  impossible.  The  value  of  K,  i.e.,  the  radius  of  the  darker  body  cannot,  therefore, 
have  either  of  these  latter  values. 

Substituting  the  values  of  p  for  the  first  three  assumptions  for  K  in  equation  (15)  above, 
we  shall  have  the  following  15  observation  equations : 

OBSERVATION  EQUATIONS. 


(e  =  0.80 

K  =  085 

K=1.00 

r  —  yx  for 
K  =  0.80,       K  =  0.85,        K  —  1.00 

(  1)    0.7500  a;  +  0.2500  y 

=2.3725  ; 

=2.3839; 

=2.9588 

1.4915 

1.7764 

2.0207 

(  2)    0.5868     +0.4132 

=1.7625  ; 

=1.8761  ; 

=2.2249 

1.7561 

1.8052 

1.9543 

(  3)    0.4132     -t  0.5868 

=1.2071  ; 

=1.2953; 

=1.5555 

1.7281 

1.7776 

1.9270 

(  4)    0.2500     +0.7500 

=0.7123  ; 

=0.7767  ; 

=0.9553 

1.6411 

1.6878 

1.8358 

(  5)    0.1170     +0.8830 

=0.3614; 

=0.4092  ; 

=0.5271 

1.6746 

1.7274 

1.8767 

(  6)    0.0302     +0.9698 

=0.1213; 

=0.1561; 

=0.2294 

1.1693 

1.1693 

1.2300 

(  7)    0.0076     +0.9924 

=0.0923  ; 

=0.1279; 

=0.1954 

1.2670 

1.2825 

1.4142 

(  8)    0.0000     +1.0000 

=0.0807  ; 

=0.1163; 

=0.1815 

1.4545 

1.4554 

1.6264 

(  9)    0.0076     +0.9924 

=0.0962  ; 

=0.1315; 

=0.2002 

2.0176 

2.0610 

2.2410 

(10)    0.0302     +0.9698 

=0.1867; 

=0.2252  ; 

=0.3100 

2.0303 

1.0893 

2.2417 

(11)    0.1170     +0.8830 

=0.5390  ; 

=0.5957  ; 

=0.7448 

1.8277 

1.9393 

2.1018 

(12)    0.2500     +0.7500 

=0.9938  ; 

=1.0727  ; 

=1.2982 

1.6723 

1.8275 

1.9780 

(13)    0.4132     +0.5868 

=1.3879; 

=1.5655; 

=1.8670 

1.8966 

1.8992 

2.0524 

(14)    0.5868     +0.4132 

=2.0415; 

=2.1694; 

=2.5613 

— 

2.0184 

2.1772 

(15)    0.7500     +0.2500 

^^     —  ~ 

=2.9217; 

=3.4292 

10  A    STUDY   OF   THE    LIGHT   CURVE 

Each  pair  of  these  equations  furnishes  a  value  for  both  x  and  «/,  and  from  the  rcsulls 
of  their  solution  the  values  of  r  and  cos2  i  may  be  obtained  with  the  help  of  (14a).  The 
assumption  of  a  circular  form  of  the  orbit  requires  that  the  different  values  of  r  and  of 
cos2  z,  on  the  correct  hypothesis  for  K,  shall  all  be  approximately  equal.  The  values  for  r 
obtained  by  solving  (1)  and  (2),  (2)  and  (3),  (3)  and  (4),  etc.,  in  succession  for  the  various 
values  of  K  are  tabulated  in  the  last  three  columns  of  the  foregoing  table.  The  mean 
values  and  probable  errors  for  each  of  the  assumptions  for  K  are  :  for  K  =  0.80,  r  =  1.6636  ± 
0.0485  ;  for  K  =  0.85,  r  =  1.7512  ±  0.0494,  and  for  *  =  1.00,  r  =  1.9341  ±  0.0535.  The  indi- 
vidual determinations  of  cos2  i  are  not  given  here,  but  the  corresponding  means  and  probable 
errors  are,  for  the  respective  cases : 

cos2*  =  +0.0275  ±  0.0069;   =  +0.0482  ±  0.0072;    =  +0.0547  ±  0.0074. 

The  difference  of  the  probable  errors  is  not  great  in  any  case,  but  both  r  and  cos2  i  agree  in 
their  testimony  favoring  the  smallest  value  of  K  as  being  the  most  probable.  Assuming  this  value 
of  K  however,  a  physical  peculiarity,  though  not  an  impossibility,  is  met  in  the  circumstance  that 
the  most  probable  distance  of  centres  (1.6634)  is  considerably  less  than  the  sum  of  the  radii 
(=1.8),  i.  e.,  the  masses  must  interpenetrate,  and  consequently  form  a  single  body  (Poincard's 
apiod). 

The  probable  errors  not  differing  by  enough  to  enable  them  to  pronounce  with  sufficient 
emphasis  for  any  one  of  the  hypotheses,  it  seemed  desirable  to  approach  the  problem  also  in- 
directly to  see  whether  the  conclusions  will  be  the  same  as  those  given  by  this  direct  solution. 
That  the  foregoing  discussion,  however,  indicates  conclusively  that  the  correct  value  of  K  is 
smaller  than  0.85,  there  can  be  no  doubt. 

INDIRECT  SOLUTION. 

The  mode  of  procedure  here  is  to  read  from  the  light  curve  for  suitably  chosen  epochs, 
the  instantaneous  brightnesses  in  stellar  magnitudes,  to  form  the  differences  between  these 
brightnesses  and  the  maximum  brightness,  to  convert  these  differences,  by  means  of  the 
Pogson  scale,  into  their  equivalent  light  ratios,  to  compare  these  ratios  with  the  corresponding 
ratios,  computed  from  certain  assumed  elements,  and  finally,  after  finding  sufficiently  close 
approximations  to  the  correct  values  of  the  elements,  to  adjust  these  differences  in  the  sense 
computation  minus  observation,  by  the  method  of  Least  Squares. 

Letting  J'  and  J"  denote  the  instantaneous  brightnesses  in  the  neighborhood  of  Min.  I.  and 
Min.  II.  respectively,  and  M1,  H',  M",  and  If",  the  corresponding  values  of  the  M  and  //  defined 
by  equations  (12),  it  will  be  seen  by  referring  to  my  article  on  Beta  Lyrae,  in  the  January 
Astrophysical  Journal,  that 

(10       »-** 


and  hence,  there  is  an  obvious  advantage  in  adjusting  1  —  J1  and  1  — J""  instead  of  J'  and  J". 
The  former  quantities  were  therefore  used  throughout  the  reductions. 
The  equations  for  computing  M',  or  M1'  are : 
( (a)  jS  =  40°  t. 


(b)  p  =  r  \/sm2  ft  +  cos2  i  cos2  /3.    If  »  = i'  is  near  - ,  t'  is  small  and 

—  — 


(c)o  —  r  -v/sin2  8  +  in  cos2  B.     If  i'  =  o,  p  =  r  sin  /3. 

(17) 

(d)  cos  <t>  = 


1  +  p2  -  >c2 


2P 

(e)  sin  </>'  =  -  sin  <£. 
(/)  M1,  or  M"  -  <£  +  /<2  <j>"  -  K  sin  (<£  +  <£")  =  <£  +  «2  </>"  -  p  sin 


OF  THE  VARIABLE  STAR  U  PEGASI. 


11 


These,  together  with  (16),  determine  1—  «/'  and  1—  J"  from  the  light  curve.  The  value 
of  cos2  i  as  found  above,  was  small,  and  as  a  first  approximation  i  was  taken  -,  or  i'  =  90  —  t  =  0. 

To  neglect  the  effect  of  orbital  eccentricity  requires  Min.  II.  to  fall  at  the  middle  point 
of  the  period.  Disregarding  provisionally  the  slight  displacement  of  this  chief  epoch  from 
the  middle  point,  taking  ordinates  equidistant  from  Min.  I.  and  Min.  II.  before  and  after 
these  epochs,  forming  the  means  for  each  epoch  separately  and  computing  the  corresponding 
values  of  1  —  J'  and  1  —  J",  the  results  here  tabulated  were  obtained. 


t 

Minimum  T. 

Minimum  II. 

i-j'. 

\-j". 

Before. 

After. 

Mean. 

Before. 

After. 

Mean. 

1.50 

9.32 

9.34 

9.33 

9.36 

9.35 

9.35 

0.0228 

0.0000 

1.25 

9.37 

9.37 

9.37 

9.41 

9.40 

9.40 

0.0742 

0.0362 

1.00 

9.42 

9.44 

9.43 

9.50 

9.47 

9.48 

0.1330 

0.0896 

0.75 

9.51 

9.53 

9.52 

9.61 

9.55 

9.58 

0.2072 

0.1606 

0.50 

9.62 

9.64 

9.63 

9.73 

9.67 

9.70 

0.2901 

0.2380 

0.25 

9.72 

9.72 

9.72 

9.84 

9.84 

9.84 

0.3749 

0.3018 

0.00 

9.75 

9.75 

9.75 

9.90 

9.90 

9.90 

0.4084 

0.3208 

The  results  of  this  table  are  shown  graphically  on  Plate  II. 

The  values  of  M'  and  M"  for  various  assumptions  for  K  both  greater  and  less  than 
the  minimum  value  given  above  (0.7755)  were  computed  from  formula  (16).  For  values 
of  K  less  than  0.7755,  it  was  of  course  necessary  to  assume  a  flattening  of  the  discs  and 

nyi  _     ft     ft 

to  compute  X  from  the  formula  X  =   -          — ,  (<?  depending  on  the  flattening).     After  having 

computed  a  number  of  light  curves  for  various  assumptions  for  r  and  K,  it  became  evident 
that  the  discs  must  lie  extremely  close  together.  The  attempt  was  then  made  to  ascertain 
what  value  of  K  would  best  satisfy  the  observations  on  the  hypothesis  of  contact  of  discs. 
The  residuals  M0  —  Mc  =  A  If  were  formed  for  the  values  of  t  in  the  above  table  and 
compared,  that  hypothesis  furnishing  the  smallest  mean  residual,  A  M,  being  assumed  to 
lie  nearest  the  truth.  M0  was  computed  from  the  observed  values  of  J1  and  J"  by  the  aid 
of  the  formulas, 

M'  =  (1  +  K2  X)  TT  (1  -  J') ;        M"  =  (1  +  K2A)£(1-  J");        M0  =  1/2  (M'  +  M") 

A 

and  Mc  by  the  aid  of 

Me  =  <f>  +  K2  <£"  —  p'  sin  <£,          p'  =  -  p  and  -  =  A/sin8  /3  +  </  cos2  ft. 

J  J 

The  value  K  =  0.8,  involving  a  somewhat  simpler  hypothesis  than  K  =  0.75,  viz  :  q  =  1, 
was  taken  as  a  basis  for  further  experimentation,  notwithstanding  the  fact  that  the  latter 
value  of  K  gave  a  somewhat  smaller  mean  residual.  If,  moreover,  as  seems  now  probable, 
a  part  of  the  light  change  be  ascribed  to  a  flattening,  the  larger  value  of  K  will  probably  be 
nearer  the  truth.  The  distance  of  centres,  r,  was  then  assumed  to  be  1  +  1.1  K  and  1  +  0.9  K 
in  turn,  and  the  mean  residuals  computed  on  these  hypotheses.  The  results  of  these  six 
hypotheses  are  here  tabulated.  Unless  otherwise  explicitly  stated,  it  is  to  be  understood 
that  r  =  1-f  K,  that  q  =  1,  and  that  the  components  are  similar  ellipsoids  of  revolution. 
For  the  case  where  K  —  0.75,  the  smallest  possible  value  of  q  (=  1.0271)  was  used  in 
obtaining  M0. 


12 


A   STUDY    OF   THE   LIGHT    CURVE 


VALUES  OF  J/,  31,.  AND  A  M  ON  VARIOUS  HYPOTHESES  FOR 


Jt/o                        Me                       A  M 

SI0                      Ma                       A  M 

K  =  0.75,  q  =  1.0271 

K  =  0.8 

0.0514 

0.1372 

-0.0858 

0.0538 

0.1456 

-0.0918 

0.2728 

0.2926 

-0.0248 

0.2839 

0.3303 

-0.0464 

0.5612 

0.5720 

-0.0108 

0.5732 

0.6097 

-0.0265 

0.9358 

0.9189 

+  0.0169 

0.9746 

0.9810 

-0.0064 

1.3486 

1.3300 

+  0.0186 

1.3998 

1.4274 

-0.0276 

1.7259 

1.7299 

-0.0040 

1.7915 

1.8949 

-0.1034 

Mean  residual       0.0302 

Mean  residual      0.0503 

K  =  0.9 

•          K  =  0.95 

0.0586 

0.1634 

-0.1048 

0.0612 

0.1690 

-0.1078 

0.3092 

0.3703 

-0.0611 

0.3229 

0.3911 

-0.0682 

0.6349 

0.6850 

-0.0501 

0.0632 

0.7228 

-0.0596 

1.0579 

1.1051 

-0.0472 

1.1049 

1.0957 

-0.0092 

1.5242 

1.6394 

-0.1152 

1.5919 

1.7098 

-0.1179 

1.9509 

2.1904 

-0.2395 

2.0375 

2.3206 

-0.2831 

Mean  residual       0.1030 

Mean  residual       0.1076 

K  =  0.8,  r  =  1  +  0.9  it 

«  =  0.8,  r  —  1  +  1.1  K 

0.0538 

0.2116 

-0.1578 

0.0538 

0.0882 

-0.0344 

0.2839 

0.4029 

-0.1190 

0.2839 

0.2672 

+  0.0167 

0.5732 

0.6806 

-0.1074 

0.5732 

0.5179 

+  0.0553 

0.9746 

1.0313 

-0.0567 

0.9746 

0.8897 

+  0.0849 

1.3998 

1.6714 

-0.2716 

1.3998 

1.3835 

+  0.0163 

1.7915 

1.9134 

-0.1219 

1.7915 

1.8760 

-0.0845 

Mean  residual       0.1391 

Mean  residual      0.0487 

The  low  value  of  the  mean  residual  of  the  first  hypothesis,  viz. :  K  =  0.75  and  q  =  1.0271, 
renders  a  further  investigation  into  the  general  effect  of  a  flattening  desirable.  The  light 
curves  for  y  =  1.01,  1.02,  1.03,  1.04  and  1.05  were  now  computed  and  the  residuals,  AJ!/^, 
formed  with  the  proper  values  of  X,  from  X=  (TO  —  c  q)  /  (m  —  q),  with  K  =  0.7785  and  r  =  1.7814. 

VALUES  OF  AM0_C  FOR  VARIOUS  VALUES  OF  q. 


5=1.01 

q  =  1.02 

q  =  1.03 

9  =  1.04 

?  =  1.06 

-0.0835 

-0.0607 

-0.0774 

-0.0737 

-0.0705 

-0.0182 

-0.0274 

-0.0086 

-0.0017 

+  0.0046 

-0.0031 

+  0.0018 

+  0.0295 

+  0.0377 

+  0.0483 

+  0.0229 

+  0.0244 

+  0.0557 

+  0.0676 

+  0.0788 

+  0.0149 

+  0.0113 

+  0.0389 

+  0.0493 

+  0.0609 

-0.0409 

-0.0533 

-0.0391 

-0.0330 

-0.0262 

Si.  0.0306 

0.0298 

0.0415 

0.0438 

0.0482 

OF   THE   VARIABLE    STAB   U    PEGASI.  13 

These  latter  values  of  K  and  r  resulted  from  a  Least  Square  solution  of  a  set  of  observation 
equations  connecting  d  /c,  d  r,  and  d  i'  (_—  —  d  i) ,  which  gave  d  i'  =  V—  0.0096.  This  value  of  d  i' 
being  imaginary  but  small,  it  was  put  =  0.  The  residuals  for  the  five  hypotheses  are  col- 
lected in  the  table  last  preceding. 

For  the  case  in  which  q  =  1.00,  the  third  column  for  K  =  0.8  above  may  be  examined. 

Both  the  run  of  the  individual  values  of  A  M0_e  and  the  magnitude  of  the  mean  residual 
indicate  q  =  1.02  to  be  most  approximate. 

Differential  equations  were  now  derived  in  such  form  as  to  connect  dk,  dr,  dq,  and  di'z, 
with  dM—^.M0_,..  The  derivation  of  these  relations  was  made  as  follows: 

Differentiating  M  =  <j>  +  K2  </>"  -  P'  sin  <£,  where  P'  =  (I//)  p  and  (I//)  =  Vsin2  ft  +  q2  cos2  /?, 

I  rcdu 
we  find, 


n'2  +  K2  —  1  1  +  p'2  K2  p'2  —  1  — 

and  reducing  by  means  of  cos  <f>"  =-  — ,     cos  <£  =  —        and  cos  (<£  +  </>")  =  — 

' 


dM=  (1  -  p'  cos  <t>)  d<t>  +  K2  d<j>"  +  2  *  <£"  dK  -  sin  <f>  .  dp', 

K  cos  <j>"    ,  .  /cosrf>"      sin  d>"\   7         cos  d> 

—    —  dp'     and     dd>"  =  [  -         ---  —  )  dK  ---  —dp' 
p'  sin  </>  \p'  tan  <£         p1     )  p1  sin  <f> 


p  sin 

which  give,  after  some  simplifications, 

dM=  2  <f>"  Kd«  -  2  sin  <f>dp'. 

But  dp1  =  j.  dp+p  d  ^j,  where  p  =  r  \Xsin2/3  +  i"  cos2^  for  small  values  of  9  =  -x—  i. 

Differentiating  and  substituting  we  obtain  finally, 

(A)        dM=2K<l>»  dK-fidr-  Qdq-Sdi1* 
in  which 

p  sin  <j>  r 

Jl  =  2        ^r~  ;        Q  =  1q  pf  cos2  ft  sin  <£  ;        and  I  =  -  sin  </>  cos2  /3  esc  ft. 

Computing  the  coefficients  2  K  <£",  R,  Q,  and  Jwith  the  values  q  =  1.02,  X  =  0.7748,  «  =  0.7785, 
r  =  1.7816,  and  i'  —  o,  for  the  epochs  used  in  the  foregoing  tables,  the  following  six  observation 
equations  were  obtained  :  — 


0.9154  dk 

-  0.7517  dr 

-  0.3403  dq 

-  0.2232  di13 

-0.0607  =  0 

1.2387 

-0.8591 

-  0.6383 

-0.5389 

-0.0274  =  0 

1.5827 

-0.8616 

-0.9019 

-1.0897 

+  0.0018  =  0 

1.9702 

-0.7537 

-1.0031 

-2.0139 

+  0.0244  =  0 

2.4611 

-0.5420 

-0.8119 

-3.6442 

+  0.0113  =  0 

3.4477 

-0.2205 

-0.3762 

-6.3170 

-0.0533  =  0 

These  were  rendered  homogeneous  by  putting 

dk  =  0.2901*;  dr  =  1.1606  y;  dq  =  0.9969s;  di'a  =  0.1583  w,  and  v  =  0.0607. 
The  equations  then  were  :  — 


0.2655  as 

-  0.8724  y 

-0.33923 

-  0.0353  w 

-  1.0000  v  =  0 

0.3593 

-0.9971 

-0.6363 

-0.0855 

-0.4514     =  0 

0.4591 

-1.0000 

-0.8991 

-0.1725 

+  0.0297     =  0 

0.5715 

-0.8747 

-1.0000 

-0.3188 

+  0.4020     =  0 

0.7138 

-0.6290 

-0.8094 

-0.5769 

+  0.1862     =  0 

1.0000 

-0.2529 

-0.3750 

-1.0000 

-0.8781     =  0 

14  A   STUDY    OF   THE   LIGHT    CURVE 

These  resulted  in  the  following  Normal  Equations: 

+  2.2467*  -  2.2508  y  -2.2557s  -1.7l33«o  -  0.9296  i/=0 

-2.2508  +3.9800  +3.3081  +1.1833  +1.04(32    =  0 

-2.2557  +3.3081  +3.1241  +1.3822  +0.3763     =  0 

-1.7133  +1.1833  +1.3822  +1.4727  +0.7113     =0 

The  solution  of  these  normals  gave 

x  =  -0.302;  y  =   +0.056;  z  =  -0.183;  and  w  =  -0.194 
and  hence, 

dx  =  -0.087;  dr  =  +0.064;  dq  =  -0.183;  and  di'2  =  in  =  -0.031. 

Inasmuch  as  an  imaginary  value  of  i1  (=  d  i')  can  have  no  physical  significance,  the  d  i'2  can 
only  be  put  equal  to  zero,  and  the  equations  solved  on  this  hypothesis  gave, 

d«  =  -0.088;  dr  =  -0.125;  and  dq  =  -0.170. 

The  assumed  value  of  q  was  1.02,  and  consequently  the  maximum  allowable  negative  value 
for  d  q  =  —  0.02,  a  magnitude  considerably  smaller  than  that  resulting  from  the  Least  Square 
solution. 

Another  important  contravention  of  the  physical  conditions  involved  in  the  problem  is  that 
die  and  dq  should  be  of  unlike  signs.  This  can  be  readily  shown.  We  have  seen  above  that 

Ka  >  m~g.     Taking  the  least  value  of  *2  consistent  with  physical  conditions,  viz.,  «2  = , 

cq                                             dJ               m  C3 

and  differentiating  it,  we  obtain      -  =  — .  an  essentially  negative  magnitude.  The  latter 

d  q  2ci<q2 

relationship  would  be  emphasized  more  strongly  by  using  KZ  greater  than  this  least  value. 

While,  therefore,  a  Least  Square  adjustment  can  add  nothing  to  the  accuracy  of  the 
values  obtained  experimentally,  the  magnitudes  and  signs  of  the  values  of  d  K  and  d  r  furnished 
by  the  adjustment  indicate  that  K  and  r  should  be  corrected  toward  the  values  of  these 
quantities  derived  in  the  direct  solution  of  the  first  part  of  this  paper.  Inasmuch  as  the 
foregoing  results  are  the  best  that  could  be  obtained  after  having  computed  twenty-five  or 
thirty  different  light  curves  in  which  the  elements  were  shifted  in  almost  every  conceivable 
way,  it  may  be  asserted  with  confidence  that  the  following  results  may  be  regarded  as  the 
best  attainable  in  the  present  state  of  the  observational  material : 

1.  The  light  curve  of  U  Pegasi  given  in  Harvard  College  Observatory  Circular,  No.  23, 
is  satisfactorily  represented  by  the  satellite  theory. 

2.  The  distance  of  centres  does  not  materially  differ  from  the  sum  of  the  radii  of  the 
components,  suggesting  the  probable  concrete  existence  of  the  "  apiodal "  form  of  Poincare". 

«S.  The  smaller  companion  is  about  0.77  as  bright  as  the  larger,  and  the  ratio  of  radii  is 
approximately  1:0.78. 

4.  The  inclination  of  the  orbit  is  very  nearly  90°,  and  the  disc  of  one  or  both  bodies,  if 
separate,  is  slightly  flattened. 

5.  The  accuracy  of  present  observations  does  not  suffice  to  determine  the  elements  of 
the  "  system  "  completely,  since  the  foregoing  discussion  shows  the  residuals  to  be  incapable 
of  adjustment  by  Least  Squares. 

6.  The  manner  of  rise  and  fall  of  the  observed  curve  after  and  before  the  minima,  which 
portions  of  the  curve  were  determined  with  especial  care,  fails  to  confirm  one's  first  impres- 
sion on  examining  the  curve,  viz. :  that  the  components  are  separated  enough  to  remain  apart 
for  an  appreciable  time  at  the  maxima.     The  difference  between  the  durations  of  uniform 
brightness  at  the  maxima,  as  shown  by  the  curve,  would  seem  to  indicate  a  considerable  orbital 
eccentricity,  whereas  the  small  distance  of  centres  nullifies  the  possibility  of  its  existence.     It, 
therefore,  seems  desirable  to  direct  attention  to  the  importance  of  a  careful  photometric  study 


OF   THE    VARIABLE   STAR   U   PEGASI. 


15 


of  U  Pcgasi's  light  curve  near  the  maxima,  with  a  view  to  ascertaining  whether  or  not  the 
form  of  this  curve  near  these  epochs  is  real. 

The  appended  plates  will  assist  in  forming  a  quick  judgment  of  the  degree  of  approxima- 
tion of  the  theoretical  to  the  observed  curve. 

Plate  I.  gives  the  points  of  Pickering's  curve,  the  continuous  curve  drawn  through  these 
points  and  used  as  the  basis  of  the  present  discussion,  together  with  the  computed  points 
(marked  with  circles)  of  the  theoretical  curve.  The  position  of  the  points  of  the  derived 
curve  would  conform  much  more  closely  to  the  curve  of  observations,  by  shifting  the  entire 
observed  curve  before  and  after  Min.  II.  forward  by  about  1.20  minutes,  which,  in  view  of 
the  short  period  of  time  over  which  the  observations  on  which  the  constants  of  the  equation 
of  the  light  changes  depend,  would  be  allowable.  Plate  II.  shows  the  effect  of  this  slight 
shift.  This,  of  course,  amounts  to  assuming  that  Min.  II.  lies  midway  of  the  period,  and 
yet,  since  especial  attention  was  directed  to  the  study  of  the  light  change  in  this  vicinity, 
it  does  not  seem  that  the  difficulty  would  be  likely  to  lie  here.  From  private  conversation 
with  Professor  Pickering,  I  learn  that  the  scarcity  of  observations  at  command  and  the 
shortness  of  the  interval  over  which  his  available  observations  were  distributed  made  a  definite 
determination  of  the  first  constant  of  the  equation  of  the  light  variation  of  this  star  impossible, 
and  that  only  the  third  decimal  of  a  day  can  be  relied  on.  This  suggests  the  removal  of  the 
difficulty  by  shifting  the  entire  computed  curve  forward,  or,  what  amounts  to  the  same  thing, 
the  entire  theoretical  curve  backward  by  the  above  mentioned  amount,  and  this  gives  a  wholly 
satisfactory  accord  of  theory  and  observation  for  the  entire  curve  save  at  the  maxima. 

Plate  III.  accordingly  represents  the  observed  curve  in  full  line,  the  derived  curve  in  dotted 
line,  and  the  latter,  after  the  shift  referred  to,  in  a  long  dash  followed  by  two  shorter  ones.  The 
computed  points,  enclosed  in  circles,  are  also  given  in  their  true  (unshifted)  positions. 

Barring  the  vicinity  of  the  maxima,  for  which  further  observations  must  be  awaited,  the  rep- 
resentation may,  the  writer  thinks,  be  regarded  as  provisionally  satisfactory,  and  that  U  Pegasi 
is  to  be  regarded  as  varying  by  reason  of  the  mutual  occultations  of  revolving  components. 

The  following  table  contains  for  corresponding  epochs  the  grade  values  of  the  ordinates 
of  both  the  computed  and  observed  curves,  together  with  the  residuals  in  the  sense  observa- 
tion —  computation  for  the  final  unshifted  curve.  Aside  from  the  fact  that  the  errors  are 
systematic,  i.  e.,  all  of  same  sign  (which  are  almost  wholly  gotten  rid  of  by  the  aforesaid 
shift),  but  little  more  could  be  desired,  and  the  exceedingly  small  values  of  the  average 
deviations  deprives  the  residuals  of  almost  all  significance.  Applying  the  mean  residual  as 
a  correction  to  the  individual  residuals,  which  is  the  same  as  adopting  the  shifted  curve  as 
final,  the  representation  becomes  entirely  satisfactory. 

COMPARISON  OF  COMPUTED   WITH  OBSERVED   CURVE. 


t 

'. 

>. 

"_ 

•To 

/. 

"U 

1.50 

m. 

9.35 

m. 

9.36 

m. 

-0.01 

m. 

9.34 

m. 

9.36 

m. 

-0.02 

1.25 

9.41 

9.41 

0.00 

9.37 

9.39 

-0.02 

1.00 

9.48 

9.48 

0.00 

9.43 

9.45 

-0.02 

.75 

9.58 

9.58 

0.00 

9.52 

9.52 

0.00 

.50 

9.70 

9.72 

0.02 

9.62 

9.62 

0.00 

.25 

9.84 

9.87 

0.03 

9.72 

9.73 

-0.01 

.00 

9.90 

9.90 

0.00 

9.75 

9.75 

0.00 

Average  Deviations  =  0.009                                     =  0.01 

16 


A   STUDY   OF   THE   LIGHT   CURVE    OF   THE   VARIABLE    STAR   U   PEGASI. 


Figure  4  illustrates  the  geometrical  relations  prevailing  in  the  system,  on  the  hypothesis 
of  separation  of  discs.  The  resemblance  to  /3  Lyrac  is  quite  apparent,  though  there  is  an 
essential  difference  in  that,  with  the  latter  star,  the  smaller  component  is  the  brighter,  while 
with  U  Pegasi  the  reverse  is  the  case. 


FIG.  4.  — THE  SYSTKM  OF  U  PEGASI. 


In  conclusion,  the  writer  would  thank  Dean  Ricker  of  this  University  and  Professor 
Pickering  of  Harvard  College  Observatory  for  valuable  assistance  rendered  during  the  prose- 
cution of  this  inquiry :  the  former,  by  the  loan  of  a  computing  machine,  without  which  the 
laborious  computations  involved  in  this  paper  could  hardly  have  been  made  during  the  progress 
of  regular  University  work;  and  the  latter,  by  granting  the  writer  every  possible  means  of 
acquainting  himself  personally  with  the  working  methods  and  of  forming  an  idea  of  the 
attainable  accuracy  of  the  polarizing  photometer. 

CAMBRIDGE,  MASS.,  August,  1808. 


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